The exact choice of wording should not be given too much weight.
However, we feel that any notion of power in the `hybrid' Bayesian/frequentist setting should be conditional on a relevant effect (or at least a non-null effect) to preserve the conditional nature of the purely frequentist power.
Using the term `power' to refer to a joint probability like
the `expected power' of
[1]} and [2]} (our \(\operatorname{PoS}(n)\) ) or
the `average/expected power' of
[3]} (our \(\operatorname{PoS}^{\prime }(n)\) ) is potentially misleading.
Others suggest `conditional expected power' for \(\operatorname{EP}(n)\)  to distinguish it from `expected power' (our \(\operatorname{PoS}^{\prime }(n)\) ) [1]}, [2]}.
This wording, however, may lead to confusion when also considering interim analyses where `conditional power' is a well-established term for the probability of rejecting the null hypothesis given \(\theta _{{\operatorname{alt}}{3.5pt}}\)  and partially observed data [6]}.
